Calderón-Zygmund operators and endpoint spaces for Hermite expansions

Abstract: Let $L=-\Delta +|x|^2$ be the Hermite operator on $\mathbb{R}^n$, and $T$ be a Calder\'on-Zygmund type operator that is modelled on certain singular integrals related to $L$. We establish necessary and sufficient conditions for $T$ to be bounded on various function spaces including the Hardy spaces and the Lipschitz spaces associated to $L$. We then apply our results to study the boundedness of the Riesz transforms and pseudo-multipliers associated to $L$.